Question: $\dfrac{ -2v + 2w }{ 5 } = \dfrac{ 5v - 2x }{ 4 }$ Solve for $v$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -2v + 2w }{ {5} } = \dfrac{ 5v - 2x }{ 4 }$ ${5} \cdot \dfrac{ -2v + 2w }{ {5} } = {5} \cdot \dfrac{ 5v - 2x }{ 4 }$ $-2v + 2w = {5} \cdot \dfrac { 5v - 2x }{ 4 }$ Multiply both sides by the right denominator. $-2v + 2w = 5 \cdot \dfrac{ 5v - 2x }{ {4} }$ ${4} \cdot \left( -2v + 2w \right) = {4} \cdot 5 \cdot \dfrac{ 5v - 2x }{ {4} }$ ${4} \cdot \left( -2v + 2w \right) = 5 \cdot \left( 5v - 2x \right)$ Distribute both sides ${4} \cdot \left( -2v + 2w \right) = {5} \cdot \left( 5v - 2x \right)$ $-{8}v + {8}w = {25}v - {10}x$ Combine $v$ terms on the left. $-{8v} + 8w = {25v} - 10x$ $-{33v} + 8w = -10x$ Move the $w$ term to the right. $-33v + {8w} = -10x$ $-33v = -10x - {8w}$ Isolate $v$ by dividing both sides by its coefficient. $-{33}v = -10x - 8w$ $v = \dfrac{ -10x - 8w }{ -{33} }$ Swap signs so the denominator isn't negative. $v = \dfrac{ {10}x + {8}w }{ {33} }$